Brainball

The following images were copied as the site http://www.brainball.de is not necessarily available.

The toy consists of a code wheel, that can be rotated as whole around the sphere, as well
as paired sectors can be flipped.

definitions

Be a shift clockwise regarded as positive, and a shift counter-clockwise be regarded
as negative. Without restricting any solutions, be the shifts within -6 and 7.
A rotation is a half turn on the axis, while holding the center piece. Be the
the axis horizontal and the movable part with 4 numbers to the left. Counting the
positions starts on the center piece top left, clockwise.
Be an operation defined as a shift followed by a rotation.
Be a sequence a number of operations followed by a shift.
Be a setting of numbers a vector, the startup vector is 1,2,3,4,..13, yellow.

Points of interest

The space of possibilities is said to be 13! * 2^13 = 51 10^12. Opon some quick trials,
I found several cycles, paths giving the identity. I also found several paths leading to
the same combination.
Hence the questions :

from a given vector, are all possible vectors reachable ?

how long is the longest of the shortest paths ?

Exhaustive search requires some acceleration. Each operation has 13 choices, meaning
with

4 operations, there are 28k choices

6 operations, there are 4.8M choices

8 operations, there are 815M choices

10 operations, there are 138G choices

As newbie I guess a research on the cycles looks promising.

Cycles

Be a cycle the N fold repetition of an operation or the N fold repetition of multiple
operations leading to the start vector.

operation(s)

times for identity

+1/ ( or -1/ symmetry)

8

+2/

12

+3/

4

+4/

12

+5/

6

+6/

14

+7/

14

+1/-1/

20

+2/-2/

12

+1/+2/ ( or +2/+1/ commute)

13

+1/+2/+3 or any permutation

12

+2/+2/ as expected

6

+3/+3/ as expected

2

+3/+3/+3/

4

+3/+3/+3/+3/ as expected

1

+1/+2/+3/+4/

16

+1/+3/+2/+4/

40

+2/-2/+1/

72

Due to symmetry, changing the sign of the shift(s) gives the same result.
There is a great number of short cycles. To fill the space of paths, they have
to be avoided.
It appears strange that unsystematic tests with up to 7 operation to be repeated
showed rather short cycles.

short paths

It was somewhat apparent that multiple path from one vector lead to the same vector.
There seem to be a greater number than assumed. Some unsystematic tests with the
number of path found depending on the search level (= how many operations)

operations

3

4

5

6

7

+1/

3

3

67

67

1443

-1/

3

3

67

67

1423

+2/

3

3

67

67

1422

-2/

3

3

67

67

1417

+1/-1/

1

5

5

114

NA

-1/+1/

1

5

5

115

115

As can be seen, the symmetry is gone, and some searchlevels add
nothing.

To be continued ...

software

finding a path between start and end vector

recursively do a shift and a rotation, then check whether any shift
would match the end vector. Let the shifts run from -6 to +7 and
go to a predefined depth.
I may publish the software at a later stage on one of my delphi pages